张衡的太阳运动算法及其解释

The Algorithm and Explanation of the Zhang Heng’s Solar Motion

东汉张衡作《浑仪图注》对浑仪原理和使用方法进行了注解。贾逵认为“日行”(太阳运动)在黄道上均速移动,但史官用赤道度来测量却显示太阳运动不均匀。张衡认为将赤道度变换成黄道度就会使“日行”回归平均运动,不会有进退。为此他紧贴浑天仪的表面用细篾条贯穿两极,测量黄道度与赤道度的对应值并进行了数值处理。他设置黄赤道差变化率相同而距度不等的“同率”区间,将这些微小区间的差分累积起来,得到节点上的黄道度和赤道度,再用线性内插法求节点之间的数值。张衡的方法类似现代数学中的不等距“样条插值”与“数值积分”方法,是中外数学史上的重大创举。

Zhang Heng in the Eastern Han Dynasty annotated the principle and usage of the armillary sphere in his Notes to the Armillary Sphere Diagram.Jia Kui believed that the“solar movement”(solar motion)at a uniform speed on the ecliptic,but the historiographer's measurements of the equatorial degrees show that the solar motion is non-uniform.Zhang Heng believed that transforming the equatorial degrees into the corresponding ecliptic degrees to measure solar motion would bring the solar motion back to its average motion,without any advance or retreat.For this reason,he used thin bamboo strips to appress the surface of the armillary sphere and connect the two poles,measured the corresponding values of the ecliptic and equatorial degrees,and then conducted numerical processing on the measured data.He set up the“identical rate”interval with the same rate of change of the difference between the ecliptic degrees and the equatorial degrees,but different steps in length.He then accumulated the differences between these small intervals to obtain the ecliptic and equatorial degrees on the nodes,and used the linear interpolation method to calculate the numerical value between the nodes.Zhang Heng's method is similar to the unequally spaced“spline interpolation”and“numerical integration”methods in modern mathematics,and is a significant innovation in the history of mathematics both in China and abroad.